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Properties

Label 2-1840-460.459-c1-0-30
Degree $2$
Conductor $1840$
Sign $-0.0789 - 0.996i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.49·3-s + (0.689 + 2.12i)5-s + 4.18i·7-s − 0.754·9-s + 6.19·11-s + 0.773i·13-s + (1.03 + 3.18i)15-s + 0.0856·17-s − 4.78·19-s + 6.27i·21-s + (3.29 − 3.48i)23-s + (−4.04 + 2.93i)25-s − 5.62·27-s + 6.23·29-s + 1.76i·31-s + ⋯
L(s)  = 1  + 0.865·3-s + (0.308 + 0.951i)5-s + 1.58i·7-s − 0.251·9-s + 1.86·11-s + 0.214i·13-s + (0.266 + 0.823i)15-s + 0.0207·17-s − 1.09·19-s + 1.36i·21-s + (0.687 − 0.726i)23-s + (−0.809 + 0.586i)25-s − 1.08·27-s + 1.15·29-s + 0.316i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0789 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.0789 - 0.996i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -0.0789 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.494236969\)
\(L(\frac12)\) \(\approx\) \(2.494236969\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.689 - 2.12i)T \)
23 \( 1 + (-3.29 + 3.48i)T \)
good3 \( 1 - 1.49T + 3T^{2} \)
7 \( 1 - 4.18iT - 7T^{2} \)
11 \( 1 - 6.19T + 11T^{2} \)
13 \( 1 - 0.773iT - 13T^{2} \)
17 \( 1 - 0.0856T + 17T^{2} \)
19 \( 1 + 4.78T + 19T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 - 1.76iT - 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 + 6.59T + 41T^{2} \)
43 \( 1 - 1.54iT - 43T^{2} \)
47 \( 1 + 9.34T + 47T^{2} \)
53 \( 1 - 8.12T + 53T^{2} \)
59 \( 1 + 12.7iT - 59T^{2} \)
61 \( 1 - 0.753iT - 61T^{2} \)
67 \( 1 - 3.43iT - 67T^{2} \)
71 \( 1 + 5.70iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 + 0.211iT - 89T^{2} \)
97 \( 1 + 5.35T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.343617263766492777634998643195, −8.616137379928736079399870999914, −8.294323833021985210885933910823, −6.79485148442043530719249836193, −6.47077046391918184151758300381, −5.62203106500523144844015653403, −4.36467092838363864421912437530, −3.29979576716318708708944233717, −2.60550485891070454245400195757, −1.79591221223438848320960269285, 0.846968869138372281535148740490, 1.76818887157392867834219288739, 3.24153497195395669187878574021, 4.07449913137029527685866538201, 4.61282749005462210197422747105, 5.95046017441477205418326717973, 6.73892303533406335760631268265, 7.57214083241358385017698572583, 8.467735360159956874972110864048, 8.914568152677128297770913045626

Graph of the $Z$-function along the critical line